Beth Number

\beth_0=\aleph_0

be the cardinality of any Countably Infinite Set ; for concreteness, take the set

\mathbb{N}

of Natural Number s to be a typical case. Denote by ''P''(''A'') the Power Set of ''A'', i.e., the set of all subsets of ''A''. Then define

:

\beth_{\kappa+1}=2^{\beth_\kappa},

which is the cardinality of the power set of ''A'' if

\beth_\kappa

is the cardinality of ''A''.

Then

:

\beth_0,\ \beth_1,\ \beth_2,\ \beth_3,\ \dots

are respectively the cardinalities of

:

\mathbb{N},\ P(\mathbb{N}),\ P(P(\mathbb{N})),\ P(P(P(\mathbb{N}))),\ \dots.

One can also show that the Von Neumann Universe s

V_{\omega+\alpha}

have cardinality

\beth_\alpha

.

Each set in this sequence has cardinality strictly greater than the one preceding it, because of Cantor's Theorem . Note that the 1st Beth Number

\beth_1

is equal to ''c'' (or

\mathfrak c

), the Cardinality Of The Continuum , and the 2nd Beth Number

\beth_2

is the cardinality of the power set of the continuum.

For infinite Limit Ordinal s κ, we define

:

\beth_\kappa=\sup\{\,\beth_\lambda:\lambda<\kappa\,\}

.

If we assume the Axiom Of Choice , then infinite cardinalities are linearly ordered; no two cardinalities can fail to be comparable, and so, since by definition no infinite cardinalities are between

\aleph_0

and

\aleph_1

, the celebrated Continuum Hypothesis can be stated in this notation by saying

:

\beth_1=\aleph_1.

The Generalized Continuum Hypothesis says the sequence of beth numbers thus defined is the same as the sequence of Aleph Number s.

The more general symbol

\beth_\kappa(\alpha)

, for ordinals κ and cardinals α, is occasionally used. It is defined by
:

\beth_0(\alpha)=\alpha

\beth_{\kappa+1}(\alpha)=2^{\beth_\kappa(\alpha)},

\beth_\kappa(\alpha)=\sup\{\,\beth_\lambda(\alpha):\lambda<\kappa\,\}

if κ is a limit ordinal.

So

\beth_\kappa=\beth_\kappa(\aleph_0).

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